If `A B=A` and `B A=B` , then show that `A^2=A` , `B^2=B` .

Options

A. `A^2B=A^2`
B. `B^2A=B^2`
C. ABA=A
D. BAB=B

Answer

To prove that \( A^2 = A \) and \( B^2 = B \) given that \( AB = A \) and \( BA = B \), we can follow these steps:

### Step 1: Start with the equation \( AB = A \)
Given:
\[ AB = A \quad \text{(1)} \]

### Step 2: Multiply both sides of equation (1) by \( A \)
We multiply both sides of the equation \( AB = A \) by \( A \):
\[ ABA = AA \]

### Step 3: Substitute \( AB \) from equation (1)
Since \( AB = A \), we can substitute \( A \) for \( AB \):
\[ AA = A \]

### Step 4: Conclude that \( A^2 = A \)
This simplifies to:
\[ A^2 = A \]

### Step 5: Now, consider the equation \( BA = B \)
Given:
\[ BA = B \quad \text{(2)} \]

### Step 6: Multiply both sides of equation (2) by \( B \)
We multiply both sides of the equation \( BA = B \) by \( B \):
\[ BAB = BB \]

### Step 7: Substitute \( BA \) from equation (2)
Since \( BA = B \), we can substitute \( B \) for \( BA \):
\[ BB = B \]

### Step 8: Conclude that \( B^2 = B \)
This simplifies to:
\[ B^2 = B \]

### Final Conclusion
We have shown that:
\[ A^2 = A \quad \text{and} \quad B^2 = B \]